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QUESTION-039: What sets the order and the moment, in training time or in examples, at which a network acquires a structure, and is acquisition one kind of event or several?

5 min read

Properties7
StatusOpen
Version2
Tagslearning-theory, representation-learning, compositionality
Linelearned-structure
Refined byQUESTION-081 — When the relative scale of a network's layers sets its learning regime under gradient flow, does that regime survive SGD, whose minibatch noise drives the relative scale toward balance, and on what timescale compared with learning?, QUESTION-158 — Does gradient descent on a two-layer network, without a spectral initialization or a data split, recover direction k of a scale-free hierarchical multi-index target at n = Θ(a_k^(−2) d) samples and attain the optimal weighted error, and do the same thresholds hold for an input law that is not isotropic Gaussian?, QUESTION-166 — Does the depth law of deep linear networks hold from starts off the decoupled set: learning time, counted in iterations at the stable step size, bounded in depth when every mode's initial end-to-end gain is of order one, and growing as a power of 1/ε when it is ε, for random orthogonal or small random weights not aligned with the task's singular vectors?, QUESTION-176 — Do networks trained by gradient descent on a mixture of random facts and a second domain reach the capacity-optimal allocation, so that their threshold size and mixing ratio are the optimal learner's?, QUESTION-175 — From generic small initialization, does gradient flow on a symmetrized Barlow Twins loss with a linear or fixed-kernel model follow the aligned trajectory, learning the top eigenvectors one at a time in order of eigenvalue?, QUESTION-193 — Does sign descent provably delay each weaker skill until the stronger one is learned when a loss's eigendirections are misaligned with the coordinates, and does changing only the optimizer turn a trained network's simultaneous skill onsets into a sequence?, QUESTION-275 — Can the development of a transformer's circuits in training be described as organizational regimes forming over shared, overlapping latent features, so that a model coupling feature allocation to circuit interaction, fitted to early checkpoints, predicts the effects of held-out training interventions that feature-only and circuit-only descriptions fitted to the same checkpoints miss?
BearingTHEORY-521 — In the thermodynamic limit, with examples and capacity growing together at fixed load, generalization error can jump discontinuously with the load when the entropy of hypotheses just above the best error vanishes faster than the training-error energy grows: a spherical perceptron's error falls smoothly as 1/α, an Ising perceptron's drops to zero at a critical load, THEORY-481 — Under gradient flow averaged over tasks and data, a linear-attention layer trained on in-context regression learns as a two-layer linear network from an aligned start does: when tasks share the input covariance's eigenbasis and the weights start and stay aligned with it, then in the long-context limit each input mode is learned on its own logistic curve whose timescale falls with the square of its variance, so learning is staged by the input spectrum, and the layer converges to the estimator preconditioned by the inverse input covariance; at finite context length the modes decouple exactly only when inputs are isotropic and the modes start equal, and otherwise couple at order 1/N, THEORY-279 — Early in training, image classifiers and language models act on the low-order statistics of their inputs, first-order before second-order, and later lose that sensitivity as they come to use higher-order structure, THEORY-212 — In a two-layer linear network trained by gradient flow, the relative scale of the two layers, which gradient flow conserves, sets the learning regime: balanced layers learn in sigmoidal stages and both build task-specific representations, while strongly unbalanced layers learn exponentially with a static kernel and leave the larger layer task-agnostic, THEORY-039 — The later phases reported in neural-network training differ in kind: after the transition a network may simplify its weights, acquire item-specific information, lose generalisation, confine its tangent kernel or saturate its units, and none of these is a measured compression of I(X;T), THEORY-200 — On sparse hierarchical data, deep networks become insensitive to swapping synonymous parts and to small displacements of informative features at the training-set size at which they learn the task, THEORY-973 — In small attention-only transformer language models of two to six layers, in-context learning, measured as the loss at token 500 minus the loss at token 50, appears abruptly in one early window of training in which induction heads form, and ablating those heads at test time removes nearly all of it; one-layer models show neither, and letting a single layer attend by the previous token brings the jump to one-layer models and moves it earlier in two-layer ones, THEORY-293 — For an optimal capacity-bounded learner on an orthogonal mixture of equally frequent random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio and in full once capacity exceeds it by the facts' total entropy, and Pythia models trained on synthetic biographies mixed into web text show a threshold in model size and in mixing ratio, THEORY-283 — For a hierarchical multi-index target with isotropic Gaussian inputs and even, separable links, direction k becomes weakly recoverable by the Bayes-optimal estimator at a sample count proportional to its squared weight to the power minus one, so features emerge one at a time through sharp spectral transitions in the number of samples; a target-agnostic spectral estimator attains the thresholds and the optimal weighted error, and a two-layer network whose first layer is that estimator, with a ridge readout, attains them in excess risk, THEORY-253 — In Liu, Liu, Michaud, Gore and Tegmark's Geometry model of independent skills with orthogonal task vectors, simulation shows skills progressing together under gradient descent, the weaker taking about 1/p of the time, and in sequence under sign descent, a weaker skill progressing fast only once a stronger is done and the ratio of their times approaching 2; on a rotated quadratic the sequence appears under sign descent only when the loss's eigendirections are misaligned with the coordinates, THEORY-229 — For a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from a small initialization of common scale aligned with the top eigenvectors of the pairs' symmetrized cross-correlation learns them one at a time, in order of eigenvalue, and ends at a top spectral solution, which is exactly a minimum-norm zero-loss solution: kernel PCA on K_Γ, THEORY-204 — On TinyStories and WikiText-103, the data-limited exponent of transformer test loss, fitted to the lower envelope of learning curves across context lengths at up to about 10^8 training tokens, overlaps γ/(2β), where β is the decay with lag of the top singular value of the token–token covariance and γ the decay of next-token conditional entropy with context length; under assumed per-lag scaling forms, a horizon n*(P) ~ P^(1/(2β)) gives the exponent min(δ, γ/(2β)), δ being the decay rate of the within-horizon excess losses, THEORY-187 — In a deep linear network trained by gradient descent on whitened inputs from decoupled, balanced initial weights, depth slows learning through the initial end-to-end strength of each mode: counting iterations as learning time over the stable step size, from strength of order one the number of iterations stays bounded as depth grows, once the step size falls with depth as stability requires, while from small strength ε the plateau lengthens from ln(1/ε) with one hidden layer towards 1/ε at great depth, THEORY-086 — In the kernel regime gradient descent fits the target eigenspace by eigenspace of the neural tangent kernel, at rates set by their eigenvalues; on uniform spherical data the eigenspaces are the harmonic degrees, and for fully connected ReLU networks the non-zero eigenvalues decay as k^(−d) at every depth
Cited as “refines” byQUESTION-305 — From small random weights not aligned with the data's eigenbasis, does a linear-attention layer trained on in-context regression come into alignment with that basis before or while its modes are learned, so that its learning is staged by the input spectrum without the alignment being assumed?, QUESTION-363 — When does the order in which stochastic gradient descent comes to occupy a network's organizations over training time follow the order in which they come to dominate the Bayesian posterior as the sample grows, and what fixes the correspondence where it holds: an effective temperature set by learning rate and batch size, the accessibility of regions and the barriers between them, or the time allowed to equilibrate?

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The question

Networks acquire structure in an order and at thresholds. These include eigenspaces fitted at different rates, plateaus that end abruptly, invariances that appear at a training-set size, and correlations resolved out to a horizon. The question is what fixes when each structure arrives, whether that is data statistics, initial scale, depth, architecture or the optimizer, and whether the transitions reported under the name of “phases” are one phenomenon.

Why the record needs it

The learned-structure line asks which structure training reaches and why. When it is reached is half of that. The record holds several mechanisms that each predict a threshold, and none of them is compared with the others. CLAIM-164 is touched: in the kernel regime the unit that gets acquired is a whole eigenspace, and the guarantee is stated only for whole eigenspaces.

What the record holds

  • THEORY-086: in the kernel regime gradient descent fits “eigenspace by eigenspace”, at rates set by the eigenvalues. Limit: linearised dynamics; Nanda et al.’s kernel moved.
  • THEORY-187: from a small start ε, depth lengthens the plateau from about ln(s/ε) towards a power of 1/ε, and costs a bounded number of iterations from an order-one start. Limit: deep linear networks, decoupled starts.
  • THEORY-200: invariance to synonyms and to displacements arrives “at the training-set size at which they learn the task”. Limit: the model builds both invariances on shared statistics.
  • THEORY-204: P tokens resolve correlations out to a horizon n*(P) ~ P^(1/(2β)), and the loss is the entropy at that horizon. Limit: it rests on assumptions; P reaches about 10
  • THEORY-039: the later phases “differ in kind”, and none is a measured compression of I(X;T). This is the caution against reading every transition as one event.
  • THEORY-229: for a symmetrized Barlow Twins loss on a linear or fixed-kernel model, gradient flow from small initialization learns the top eigenvectors of the pairs’ cross-correlation one at a time, in order of eigenvalue, each at a time set by its eigenvalue and the log of the initial scale. The self-supervised instance of the THEORY-187 mechanism. Limit: proved from aligned initialization only; stepwise growth in ResNets is an experiment.
  • THEORY-279: image classifiers and language models act on low-order statistics first, first-order before second-order, and later lose that sensitivity. Acquiring later structure can remove earlier behaviour. Limit: empirical, and the ordering is shown for first- before second-order only.
  • THEORY-253: in a toy model of independent skills, the order is frequency order whatever the optimizer, but whether onset is sequential is set by the optimizer: gradient descent learns orthogonal skills together, sign descent one after another (the Domino effect). Limit: toy-model simulation and one fitted ODE.
  • THEORY-283: for a two-layer network on a hierarchical multi-index target, direction k becomes recoverable at a sample count ∝ a_k^(−2), one at a time, through sharp spectral transitions. Limit: a theorem in sample size, not in training time; the network is a spectral first layer with a ridge readout.
  • THEORY-293: for a capacity-bounded learner on a mixture of random facts and a second domain, the facts are learned not at all below a capacity threshold set by the mixing ratio, and Pythia models show the same threshold. Limit: static, saying whether facts are learned and not when; it assumes an ideal learner.

LIT-954 files no theory and is not a single-event account: it puts thresholds and power laws in one vocabulary but treats them as two kinds of event, connectivity changes and waiting-time statistics.

What would answer it

One setting in which two of these mechanisms make different predictions about timing, measured with each work’s own quantity. For example, a hierarchical task trained from small and from order-one initial scale. Or the same skills trained by gradient descent and by sign descent: the steps of THEORY-229 are a property of gradient flow, while the Domino steps of THEORY-253 vanish under it, so swapping the optimizer separates the two mechanisms.

Note of 2026-10-26: induction heads, and several kinds of event

Three works T24 of the exchange “Experimental Infrastructure Review” drew on for QUESTION-275, which refines this one, were read on 2026-10-26.

  • THEORY-973 answers in part. In small attention-only transformers of two to six layers, the moment in-context learning arrives is the window in which induction heads form; ablating them removes nearly all of it; one-layer models show neither; and letting a single layer attend by the previous token, which removes the need for two heads to compose, brings the jump to one-layer models and moves it earlier in two-layer ones. So the moment is set, there, by when a composition becomes available. Limit: one family of models, one run per model and one data order, and a window that overlaps learning-rate warm-up. Whether induction drives the transition or accompanies a shared latent event, such as learning to compose layers, the authors leave open themselves.
  • One kind of event or several. The refined-LLC study (NOTE-1188) reports at least three per component: growth (a head’s local learning coefficient rising), refinement (it falling), and repurposing, in which first-layer n-gram heads become the induction circuit’s previous-token heads and n-grams pass between heads (THEORY-967, Proposed, from two checkpoints of one run). The per-head curves differentiate by functional type over training (THEORY-975), which bears on when heads specialize, not on when a structure is acquired, and is not listed as an answer.
  • Features. Cross-checkpoint crosscoders show the features that matter for agreement moving from token-specific to category detectors, and still shifting after accuracy has stopped improving (THEORY-1004, Proposed). Acquisition read off a task score can end before the structure stops changing.

The question is unchanged.

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